There is a very similar looking garment on page 127 of the excellent 'The World War 11 Tommy' and it is worn by a Royal Artillery gunner and is described as 'Coat, Duffel'. The fabric looks just the same but it has arms.
'Intended for use in only the coldest environments, this garment, although less restrictive than the old Tropal coat, still caused the wearer to overheat if it was worn during any strenuous activity. It was well suited to sedentary tasks or periods of inactivity
I would hazard that your waistcoat is not made of felt but of Jersey wool and was intended for use in cold environments by drivers, sentries, artillery and possibly for freezing cold desert nights! Anyway, just a theory.